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Quantum ohmic dissipation : particle in an asymmetric double-well potential
C. Aslangul, N. Pottier, D. Saint-James
To cite this version:
C. Aslangul, N. Pottier, D. Saint-James. Quantum ohmic dissipation : particle in an asymmetric double-well potential. Journal de Physique, 1986, 47 (5), pp.757-766.
�10.1051/jphys:01986004705075700�. �jpa-00210258�
757
Quantum ohmic dissipation : particle in an asymmetric double-well potential
C. Aslangul (1), N. Pottier (1) and D. Saint-James (2)
(1) Groupe de Physique des Solides de l’Ecole Normale Supérieure (*), Université Paris VII, 2, place Jussieu, 75251 Paris Cedex 05, France
(2) Laboratoire de Physique Statistique, Collège de France, 3, rue d’Ulm, 75005 Paris, France (Reçu le 21 novembre 1985, révisé le 20 janvier 1986, accepté le 23 janvier 1986)
Résumé.
-La dynamique en temps réel d’une particule dans un double puits de potentiel légèrement asymétrique
en présence de dissipation ohmique est analysée en utilisant des méthodes usuelles de mécanique statistique. A température nulle, la particule tend finalement vers le puits de potentiel le plus bas ; la relaxation a globalement un
caractère exponentiel avec éventuellement de petites oscillations; à température finie, la particule atteint une dis-
tribution d’équilibre thermique sur les deux puits de potentiel ; deux régimes exponentiels successifs sont observés.
Les résultats sur la position moyenne finale de la particule sont en accord avec le principe de bilan détaillé et sont
en fait indépendants de l’hypothèse ohmique. Pour une loi de dissipation ohmique, le phénomène de localisation
qui se produit dans le double puits de potentiel parfaitement symétrique à température nulle disparaît en présence
d’une asymétrie, bien qu’un comportement précurseur soit mis en évidence pour de très faibles valeurs de l’asymétrie.
Abstract.
2014The real-time dynamics of a particle in a slightly asymmetric double-well potential in the presence of ohmic dissipation is analysed by using standard quantum-statistical methods. At zero temperature, the particle eventually goes towards the lower potential well; the relaxation has an overall exponential character with possibly
small oscillations; at finite temperature, the particle thermalizes over the two wells; two successive exponential regimes are observed in this case. The results about the final average position of the particle agree with the detailed balance principle and are in fact independent of the ohmic assumption. For an ohmic dissipation law, the localiza- tion phenomenon which occurs in the perfectly symmetric double-well potential at zero temperature disappears
when an asymmetry is present, although a precursor behaviour is displayed for very low values of the asymmetry.
J. Physique 47 (1986) 757-766 MAl 1986,
Classification Physics Abstracts
03.65 - 05.30 -74.50 -71.50
1. Introduction.
The motion of a quantum particle in an external potential in the presence of ohmic dissipation has
been the subject of considerable recent interest. What is meant by ohmic dissipation is that the equation of
motion obeyed by the Heisenberg representation of
the particle coordinate is of the classical type
where M is the mass of the particle submitted to a potential Y(q); q is a phenomenological damping
constant or friction coefficient and F(t) is a fluctuating
force. As was emphasized by A. 0. Caldeira and A. J. Leggett [1], an equation of motion of type (1)
can be obtained for a quantum particle by coupling
it to a bath of harmonic oscillators in some prescribed
manner.
The reasons for studying equation (1) for a quantum
particle are mainly twofold. First, this equation is
believed to be approximately obeyed by the so-called
macroscopic quantum variables
-such as the phase
difference across a Josephson element in a SQUID -
and has been extensively discussed for instance by
A. J. Leggett in the context of testing the applicability
of quantum mechanics at the macroscopic level [2].
Secondly, equation (1) presents interest at a fully microscopic level, since in the classical (high tempe-
rature) limit it describes the Brownian motion of the
particle in the potential V(q) : it can thus be used as
a tool for studying quantum Brownian motion [3-5].
More precisely, several different physical problems
can be studied in the general framework of equa- tion (1) according to the peculiar type of the potential V(q), e.g. quantum free particle [3, 4], particle in a potential with a metastable minimum [1], particle in a symmetric double-well potential [6, 13] or particle
in a periodic potential [14-17]. In these last two cases,
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004705075700
it has been shown that a symmetry breaking occurs
when the strength of the coupling with the bath exceeds some threshold value, the particle then remain- ing localized in one of the wells.
The dynamics of the particle is entirely non-trivial when the temperature becomes sufficiently low. It
seems worthwhile to extend the study to other types of potentials. For example, in the present paper, our aim will be to investigate the effect of an asymmetry of a double-well potential on this dynamics. This
situation can be thought of as intermediate between the case of a particle in a symmetric double-well
potential (quantum coherence problem) and of a particle in a potential with a metastable minimum
(quantum tunnelling problem). This problem has already been taken up by H. Grabert and U. Weiss [18]
and by M. P. A. Fisher and A. T. Dorsey [19] by means
of functional integral techniques.
However, the real-time dynamics of a particle in
the presence of ohmic dissipation may be conveniently
described by using standard methods of quantum mechanics [13, 17]. In particular, such a treatment
can give in certain cases a more refined description
of the particle dynamics. Following closely the method
devised in [13], we shall use in the present paper a master equation formalism, which yields the real-time
dynamics of the particle in both situations of interme-
diate-strong coupling and of weak coupling, the expan- sion parameters being clearly different in these two situations. This paper is organized as follows : in section 2, we describe the model under study. Then,
in section 3 and 4 we successively analyse the interme- diate or strong coupling situation and the weak
coupling case. Finally, in section 5 we discuss our
results on the particle dynamics; in particular we
show that they agree with those of references [18] and [19] at large times and that they indeed allow for a more refined description of the dynamics at smaller
times.
2. The modeL
We consider a particle of mass M, moving in a slightly asymmetric double-well potential. The two minima
are located at q = ± qo/2 and are characterized by
the same small oscillations frequency Qo. They differ by a small bias energy fig and are separated by a potential barrier of height Vo. By slightly asymmetric
it is meant that;’ I B 1 hi2o. We shall limit the
study to the low temperature situation as defined by kB T min (hqog Vo). Thus the thermal activation
over the potential barrier is negligible. We shall equally suppose that this barrier is sufficiently high
and large so that the tunnelling frequency (00 of the unbiased system is much smaller than the frequency Do of small oscillations in each well. This system can be restricted to only one level in each well, and des-
cribed as usual by a spin 1/2 Hamiltonian, the operator
(1/2) qo Uz corresponding to the position operator q of the particle.
The Hamiltonian of this two-level system coupled
to a bath of harmonic oscillators can be written as :
The coupling is supposed to obey the ohmic constraint, I.e. [ 1 ] :
.p(w) represents the density of modes of the bath; the
dimensionless parameter a is related to the friction
coefficient q by
and fc( w/wc) is some cut-off function such that
fc(O) = 1, and decreasing on a frequency range of
order Wc. The cut-off frequency Wc is chosen such that Wc >> W0 and Wc >> max (kB T/h, E 1). With these
assumptions, the precise form of f. should have no bearing on the dynamics of the system in the time
domain co, t >> 1. For convenience we shall use the
exponential cut-off function introduced in [11], i.e. :
3. Intermediate or strong coupling case.
3.1 CANONICAL TRANSFORMATION OF THE HAMILTO-
NIAN.
-When the coupling with the bath is inter- mediate or strong, it is convenient to carry out a canonical transformation of the Hamiltonian by
means of the unitary operator
which diagonalizes H when both (Uo and s are equal
to zero [9, 10, 13]. The transformation S leaves (1%
invariant so that the dynamics of this operator can be studied by using the canonically transformed Hamil- tonian
instead of H [9,10,13]. fl is easily seen to be equal to
where the operators B:f: are defined by
11
Therefore, as in the symmetric double-well potential
case, the Hamiltonian (8) exhibits an interaction
759
between the system and the bath of the form
However, when an asymmetry is present, the system Hamiltonian Ns after the canonical transformation S is no more zero as it was the case in the problem of the symmetric double-well potential. This fact will be seen
to be of primary importance in all what follows.
3.2 MASTER EQUATION FOR THE REDUCED SPIN DENSITY MATRIx. - Closely following the method proposed
in [13], we shall put down the convolutionless master
equation obeyed by the reduced spin density matrix Ps(t) ?
(the trace operator acts in the bath space) ; equa- tion (11) is written at the Born approximation order
with respect to Il in - 7int flint >B ( ... >B stands
for TrB { PB... 1).,Wi’n,(- t’ is written in the interac- tion representation, both spin and bath operators evolving with their respective unperturbed Hamilto-
nians.
3. 3 SPIN DYNAMICS.
-In the ohmic dissipation case, equation (3) implies that the average values ( Bt )B, and, consequently, Hint )B’ are equal to zero. The
evolution equation for the average value ( uz(t) ),
as defined by
(the trace operator acts in the spin space), takes the
closed form :
where the relaxation functions ({J - + (t) and ({J + - (t)
are defined as :
In the presence of a non-zero bias energy ne, the spin operators (J:i: rotate with frequency s and yield
the phase factors in equations (14) so that the relaxa- tion functions qJ-+(t) and qJ+-(t) are unequal. This
results in the presence of a non-homogeneous term in the
evolution equation of O’z(t). Moreover, the pre-
sence of these phase factors will be shown to be of
primary importance in the time-behaviour of the qis.
For these reasons, the situation is fundamentally
different from that of a particle in a symmetric double-
well potential [13].
With the exponential choice of the cut-off func- tion (Eq. (5)), one gets :
with
’t" is the « temperature time », as defined by :
Note that at T = 0, exp(- A2(t)) behaves like
(wc t)- 2a at large t. It can be checked by using well-
known properties of Fourier transforms that this is
independent of the particular choice of the cut-off function.
The evolution equation (15) with the functions
A1(t) and A2(t) as given by equations (16) and (17)
is a first-order inhomogeneous differential equation
of the form :
and its formal solution can be given explicitly at any time :
Before discussing equation (20) let us study the
limit value at infinite time of a.(t) >, which can be directly deduced from the differential equation (19).
3.4 LIMIT VALUE OF uz(t).
-Let us successively analyse the cases of zero and of finite temperature.
(a) Zero-temperature case
For t » 1 e 1-1, one can write down the following
asymptotic developments of the functions f(t) and
g(t)
The limit values f(oo) and g(oo) can be exactly cal- culated ; one finds :
(T denotes the standard Euler’s gamma function).
Therefore, at zero temperature, the limit value
u.,(oo) > is equal to - sign(e), a result in accordance
with the detailed balance principle, as will be discussed later (see Sect. 5). In other words, at zero temperature, the particle always goes towa’rds the lower well at infinite time, whatever its initial location and the value of the coupling constant. This result evidently
holds as long as the preceding calculation is valid.
The range of validity will be made precise later.
One can equally remark that the first term in the
expression (20) of ( 0’:( t) > expresses the decay of the
initial value 0’:(0) >; this term dies out and the
memory of the initial value is always lost, in contrast
with what occurs in the symmetric double-well
potential [13], in which case the particle looses the
memory of its initial state only when the coupling
constant a is lower than 1 (for values of a greater than 1, the particle remains parly localized on the side where it was at time t = 0).
Note that, when the limit e --+ 0 is taken, the values of f (oo) and g(oo) tend either towards 0 (when a > 1/2)
or towards oo (when a 1/2), and thus are non- analytic with respect to s.
(b) Finite temperature case
The limit values f(oo) and g(oo) can be calculated by making use of the assumption Wc ’t 1; one finds for finite ’t
In any case, the limit value (Jz( 00) > is given by :
in accordance with the detailed balance principle.
Thus the particle eventually thermalizes, whatever
its initial position and the value of the coupling, by tending towards the thermal equilibrium state corres- ponding to the Hamiltonian Ils = (11e/2) (Jz of equa-
tion (8). This result seems strange at first sight and we
shall discuss it when investigating the validity of the
calculation.
When the limit s -+ 0 is taken, f(oo) and g(oo) as given by formulae (25) and (26) tend towards their
analogs in the symmetric double-well problem (i.e.
respectively (co’/co,) (,F7r/2) (T(a)/T(a + 1/2)) x
(ak B T/1ïQ)c)2rz-l and zero) [13]. The analyticity with respect to s is thus restored at finite temperature.
3.5 TIME-EVOLUTION OF ( uz(t». - The explicit expression of uz(t» is given by equation (20).
However, since the functions f(t) and g(t) are them-
selves defined as integrals (see Eq. (15)), a full analytical
discussion of equation (20) is not so easy. Therefore,
we will try to give when possible approximate expres- sions for ( uz(t) >. When this will be impossible, we
will resort to numerical evaluation of equation (20).
For times much shorter than s 1-1 (which is a time
characteristic of the proper evolution of the system, i.e. without the bath), the evolution of ( Uz(t) )
remains very close to its evolution in the symmetric
double-well potential [13] (see Fig. 1). In other words,
the effect of an asymmetry of the double-well potential
on the behaviour of uz(t) ) is negligible as long as t 1 I le I-,.
Let us successively discuss the cases of zero and of finite temperature.
(a) Zero-temperature case
It may be seen on equations (21) and (22) that, at
zero temperature, when
the oscillating terms can safely be neglected and the
evolution of O"z(t) > approximated by the exponen- tial law :
where the relaxation time TR is equal to (f( 00») - 1. At
zero temperature, /(oo) is given by equation (23). The
associated relaxation time has been plotted as a func-
tion of a in figure 2. It is seen that the non-standard
Fig. 1.
-Time evolution of O’%(t) > at T
=0 for times
much shorter than B ,-1. Parameters : roo/roc = 5 x 10-2;
e/wc = + 10-4.
761
Fig. 2.
-Zero-temperature relaxation time TR(a) as a
function of a. Parameters : coo/co,
=5 x 10-2 ; e/co,
+ 10-2 (curve 1) or s/co,, = + 10-’ (curve 2).
character of the relaxation already found in the sym- metric double-well potential in the presence of ohmic
dissipation does persist in the presence of an asymme-
try (at least at intermediate or strong coupling) : indeed, the stronger is the coupling to the bath, the
slower is the relaxation.
As already stated, the relaxation time TR has for
limit value as I 8 I -> 0, either 0 (when a 1/2), or
00 (when a > 1/2). This has to be related to the beha-
viour of l1z(t) ) in a symmetric double-well potential,
in which l1z(t) evolves according a generalized
Kohlrausch law, more rapidly or more slowly than exponentially, depending on whether a 1/2 or
a > 1/2 [13].
For a fixed value of I 8 t, the evolution time of
l1z(t) >, instead of being ( E 1-1, is renormalized by
a factor
which, for a > 1, becomes extremely large when I 8 I -+ 0. The spin thus displays a behaviour which is
a precursor of the transition which takes place at
a = 1 in the symmetric double-well potential.
However, it is worthwhile to emphasize that this
precursor behaviour does manifest itself only for extremely small values of the asymmetry. This is very
clearly seen numerically. For instance, for a ratio
wolwe ’" 10 2 and an asymmetry characterized by
a ratio E I/we ’" 10-1, one gets (for a - 1) : we TR - 105 ; if wc ~ 1013 s-1, which corresponds to a typical phonon frequency, this amounts to a relaxation time
TR ~ 10- 8 s. It is only for extremely small values of the asymmetry that TR would recover macroscopic
values of the order of the evolution times in the sym- metric double-well potential [13] : for instance, for an asymmetry ratio E I/wc ~ 10-6, one would get TR - 10-3S.
This can equally be seen in the following way : the influence of the asymmetry of the potential on the
behaviour of O"z(t) > begins to be effective for times
t » I s 1-1. For an asymmetry ratio 8 IIWe ’" 10-1 (and a N 1), this corresponds to times t >> 10 W; 1,
a very small value indeed; but for an asymmetry ratio [ 8 I/wc ~ 10 6, this corresponds to times t > 106 (.0; 1,
which is a large value.
This illustrates once more the fact that the transition between a localized and a delocalized state which takes place at zero temperature for a particle in a perfectly symmetric double-well potential is extremely
sensitive to small modifications of the parameters : indeed we showed in [13] that the localization pheno-
menon disappears at finite temperature; similarly,
the present calculation proves that it also disappears,
even at zero temperature, when an asymmetry is present in the potential.
In summary, the zero-temperature behaviour of
O"z(t) ) is either very close to its behaviour in the
symmetric double-well potential (for times t « I B 1- 1),
or it obeys an exponential relaxation law (Eq. (28)) (for times t > to - I B 1-1 (2 r(2 (X)ln)1/2a.).
In any case, it is possible to calculate numerically O"z(t) ) as a function of time by using equation (20).
The curves displaying O"z(t) > as a function of time for different values of a and of e are plotted on figures 3a
and 3b for zero temperature. Let us now briefly com-
ment these curves. First, as already stated, the relaxa- tion gets slower and slower when a increases. On the other hand, it is apparent that, in most cases, the relaxation of ( Oz(t) is exponential, with time
constant TR ; however, for relatively low values of I s I
and of a, small oscillations of period 2 nll 8 I may
persist around an overall exponential decay, even
in the presence of ohmic dissipation (see Figs. 3a and 3b, case a = 0.25, s = 0.1). These oscillations can be traced back to the presence of oscillating terms in f(t)
and g(t) (Eqs. (21) and (22)). Yet, the competition
between two effects (amplitude - I s I-1 and rele-
vance of the asymptotic expansion t » s -1 ) res-
tricts the observability of these oscillations to a narrow
domain in the (a, [ 8 1) plane.
(b) Finite temperature case.
At finite temperatures (see Figs. 4a and 4b), two
different regimes are observed. When t >> i/2 a, the thermalization of Oz(t) > proceeds exponentially
with a time constant T,(7) = 1/f(oo), with f(oo) as
given by formula (25). In contradistinction, when
Fig. 3.
-Time evolution of ; az(t) > at T
=0 for various values of a. Parameters : coo/co,
=5 x 10- 2 ; 8/roc = + 10-1.
(a) plots a(t) = ( O’z(t) > with O’z(O) >
=sign (s) (= 1);
(b) plots - In a(t) = - In [(1/2) (( O’z(t) > + sign (s))].
Fig. 4.
-Time evolution of ; Uz(t) > at T > 0 for various values of a. Parameters : oj,lcor
=0.05; sla),,: = + 0.1;
nk B T /1iwc
=0.1.
(a) plots a(t) ; note that, at finite temperature, the final value of Q(t) is given by - th (fJ "8/2) ;
(b) plots - In a(t).
t i12 a, the dynamics is very close to the zero-tem-
perature one since (t/,r)/sinh (t/i)
~1 (see formula (17)). In particular, when oscillations exist at T
=0, they still appear
-with a smaller amplitude
-at
finite temperature, provided that I e I - ’ i/2 a. In
the opposite case, they are completely blurred out by the thermal effects (Fig. 5).
3.6 CONDITIONS OF VALIDITY OF THE TREATMENT.
-Let us now make precise the conditions under which the convolutionless equation (15) and the results which were derived from it in the preceding para-
graphs are valid.
Following general arguments developed by van Kampen [20] and by Hashitsume et al. [21], the
convolutionless equation (15) is valid as long as two
time scales can be clearly delineated : the short one, which characterizes the relaxation kernel, and the long one, which characterizes the operator of interest
(here (Jz(t) ».
A strict delineation of the domain of validity of equation (15) would be fairly delicate since the short
Fig. 5.
-Time evolution of uit) > at T > 0 for a
=1/4, co,/co,
=0.05, e/IDe = + 0.1.
(a) plots a(t) and figure 5(b) plots - In a(t). In both figures,
curve a corresponds to nkB T /lie = 2.5 ; the zero-tempe-
rature oscillations are blurred out by thermal effects ;
curve b corresponds to nkB/lie = 0.1; the zero-tempera-
ture oscillations still appear.
763
time scale rr is linked both to the system and to the bath. For the sake of simplicity, we shall limit the dis- cussion to the zero-temperature case, for which we shall give only qualitative arguments.
The short time scale Tc characterizes the relaxation kernel which is not a simple exponential function with
a well-defined time constant (Eq. (15)).
When the coupling constant a is sufficiently high,
the function f(t) (Eq. (19)) takes its maximum value for tan-1 We t
Nn/4 a, independently of E ; in other
words, the oscillating terms are then irrelevant for
defining the short time scale ’tc ; one thus can roughly
take for ’tc the time after which the function (1 +
w; t2)-CZ has decreased by a factor lle, which yields,
as in the symmetric double-well potential case [13]
Let us emphasize that this choice of Tr cannot be valid
for too low values of a. We shall fix the boundary at
a
=1/2; indeed, above this threshold, f(oo) and g(oo)
tend towards zero with e, which ensures that the relaxation kernel decreases sufficiently quickly, even
when s = 0.
Below a = 1/2, the role played by the oscillating
terms is an essential one since they insure the conver-
gence of the integrals defining f(oo) and g(oo). Using
the asymptotic developments of f(t) and g(t), valid
for times t >> s 1-1 (Eqs. (21) and (22)), the short time scale ’tc is conveniently defined as the time after which the correction f(t) - f(oo) (resp. g(t) - g(oo)) is a
fraction lle of the final value f(oo) (resp. g(oo)) ; this
criterion yields
Note that, for a 1/2, I B I ’t’c 1, as required.
It remains to characterize the time scale of (1z(t) ).
As quoted above, even in the presence of small oscillat-
ing terms, an overall exponential decay is observed,
so that the significant long-time scale of (1z(t) ) is TR. Therefore, the criterion of validity of our calcula-
tion (separation of time scales) will be written as
with Tr given by formulae (30) or (31) depending on
whether a > 1/2 or a 1/2.
Let us first study the case a > 1/2. Condition (32)
can be written as :
where we have dropped irrelevant factors of order 1.
Clearly, when a > 1/2, the factor (wc/I s 1)2(1-1 in the
r.h.s. of equation (33) is very large, and condition (33)
is automatically satisfied, whatever the value of I c 1.
When a 1/2, condition (32) can be written as :
Clearly, in this range of values of a, the asymmetry I 8 I cannot be arbitrarily low since TR -+ 0 while
’tc -+ 00 when I c I -+ 0 for a given a. In other words,
the limit s -+ 0 cannot be taken.
At this point, a few comments are worthwhile.
(i) Naively a different result would have been
expected in connexion with the well-known behaviour in the symmetric double-well at T
=0. Indeed in this last case it has been shown that the renormalized
tunnelling frequency does vanish for a finite a (namely
a >, 1), indicating a symmetry-breaking. For a slightly asymmetric potential one could have expected a simi-
lar result with a > 1 (possibly 8-dependent), i.e. the particle would remain localized in the side of the double-well in which it was at t = 0. For a 1 in the
symmetric double-well the particle is not localized,
and we could have expected a similar result in the
asymmetric situation, the non-zero average equili-
brium position corresponding to the Hamiltonian
-
(l1wo/2) Qx + (hsl2) (lz. This is not what is found here. Indeed the particle for a > 1/2 always goes for T = 0 towards the lower side of the double-well, i.e.
cr(oo) = 2013 sign (c). In other words in this case the
renormalized tunnelling frequency never strictly
vanishes so that the particle ultimately goes down to
the lower well. This is explicitly seen by looking at the
relaxation times of both problems (e.g. here TR = ( f (oo))-’ as defined by Eq. (23)). However, for times
« I P, I - 1, as remarked above, the behaviour of the
particle is close to that of the symmetric case, i.e. for
a > 1, ( O’z(t) remains close to its initial value
ar.(O) ), while for a 1 it goes quickly to nearly
zero (see Fig. 1). Eventually, in both cases, ( O’z(t) )
tends slowly towards it final value
-sign (s). This
restores a kind of continuity with the symmetric
case which therefore retains its physical interest.
This is reminiscent of the cubic potential case as treat-
ed by A. 0. Caldeira and A. J. Leggett [ 1], who showed
that for sufficient coupling the tunnelling frequency
is reduced but that the particle always has a possibility
to cross the barrier. It seems that a similar phenome-
non occurs for a particle in a double-well as soon as the
potential is not strictly symmetric. Indeed similar results have been found by A. J. Leggett et al. [22], who
find them strange and thus question the validity of
their technique at large times. We are aware of the fact that in our case, the Bom approximation relying on a perturbation series could fail in a certain region of the (a, s) plane. Checking this point is a rather formidable task that we have not yet undertaken.
(ii) When equation (32) is not satisfied, numerical calculation shows that ( pz(t) may grow outside the interval ( - 1, + 1) despite the fact that the trace of the density matrix is conserved : this is a direct proof
of the failure of the approximation
(iii) On the other hand, when the calculation is
valid, the general behaviour of O’z(t) ) can be under-
stood as follows. Equation (20) can be rewritten as
Clearly, for CTz(O) > :F - sign (8), the second term in the r.h.s. is negligible, since it contains the factor
(a)o/co,)’ 1, multiplied by a finite small integral.
Therefore, the behaviour of CTz(t) > + sign (8), i.e.
the relaxation of the particle position towards the
lower potential well is governed by exp(
f(t")), which, as f(t) itself, contains a linear term plus
and oscillating term. This explains the overall expo- nential behaviour found above. On the contrary, for
a.(O) > = - sign (E), only the second (small) term
is left in the r.h.s. of equation (35), so that O’%(t) >
always remains very close to - sign (s).
It must be noted however that at finite temperature and for t » T/2 a, a similar argument leads to a diffe-
rent conclusion : namely in this case, the second term of the r.h.s. of equation (35) becomes essential since it ensures the thermalization of a.(t) >.
4. Weak coupling case.
As previously explained, the weak-coupling case
a 1 cannot be treated by the method used above,
and another procedure has to be followed. We shall take into account the fact that now in the Hamilto- nian (2) the interaction between the particle and the bath, as described by the term
is weak. The master equation formalism is still appli- cable, W being considered as a small perturbation.
In the absence of any coupling with the bath, the
spin oscillates with frequency
For instance, if at t = 0 the particle is localized in the right-hand well « u z(O) > = 1), one gets at time t :
The oscillation of Qz(t) > is thus centred around a
positive value, whatever the sign of 8 or, in other
words, of the asymmetry.
In the presence of the coupling, one can write down
the time-evolution equation of the reduced spin density matrix under a convolutionless form [20, 21] ;
since the Hamiltonian H as given by formula (2) is
of the type :
one gets, at the Born approximation order with res-
pect to W :
The evolution equations for the average values
ai(t) ), as defined by
take the form :
where the relaxation functions Fi(t) (i = 1, 2, 3) and
the functions ’P¡(t) (i = 1, 2) are defined as :
765
Here Nn = 1/(exp(phco,,) - 1) denotes the distribution function of phonons at thermal equilibrium at tem- perature T. One can remark that the evolution equa- tions (42) with formulae (43) are analogous to the weak- coupling results derived by D. Waxman [23] through
a different technique.
In this case of weak coupling, we do not expect peculiar effects in the relaxation process and therefore
we shall limit the discussion to a study of the limit value
of (Jz(t). A simple calculation within the Born
approximation shows that, whatever the precise dis- sipation mechanism (i.e. ohmic or not, provided that p(w) G((w)2 coth (P 1iro /2) vanishes with ro), the limit value ( (Jz( 00) ) is given by
This property in fact results from a detailed balance condition relating the functions Fi(oo) and W,(oo),
which are time-integrals of correlation functions of boson operators.
The result (44) is by no way surprising : when the coupling with the bath is weak, the average position
of the particle tends towards the thermal equilibrium
value corresponding to the particle Hamiltonian of the form - (fico’/2) (cos B Qx - sin 6 6z). In particular, at
zero temperature, the average position tends towards the value - 8/mi corresponding to the quantum
ground state. Let us recall that these results are inde-
pendent of the particular dissipation mechanism.
If the asymmetry is very weak, i.e. if I s I mo,
equation (44) shows that a.(oo) > - 0, in accordance with the result found in similar conditions (weak coupling) in a symmetric double-well potential. On the
other hand, if the asymmetry is high enough, i.e. if I i; I >> coo, equation (44) shows that a,(oo) > - -
tanh (/Me/2), so that we recover a result similar to that obtained in section 3 for intermediate or strong coupling. Let us recall that the result of section 3 is valid even when a 1/2, provided that the asym-
metry is sufficiently high (see Eq. (34)). Equation (44)
shows that this can be extended to the weak coupling
situation.
Let us now discuss the results obtained in sections 3 and 4 from an overall point of view.
5. Discussion and conclusion.
In the preceding sections, we carried out two different calculations, according to the strength of the coupling
with the bath. When the coupling is intermediate or
strong, we showed that the equilibrium value of the
particle position is equal to - tanh (phel2); moreover,
we gave in this case a description of the dynamics of
the particle. On the other hand, in the case of weak coupling, we showed that, in agreement with the detailed balance condition, the particle position tends
towards - (8/mi) tanh (Phco’12) at infinite time. This last result does not depend on the particular dissipation mechanism, provided that p(ro) I G(w) 12 coth(p1iro/2)
vanishes with o. Let us see whether a similar property holds in the case of intermediate or strong coupling.
5.1 DETAILED BALANCE PRINCIPLE.
-When the
coupling is intermediate or strong, the dynamics of the particle position is most conveniently described by using the canonically transformed Hamiltonian H instead of H. We shall assume that the dissipation is
such that
where the exponent 6 reduces to 1 in the ohmic dis-
sipation case. As already indicated, when the ohmic constraint (3) is imposed, the average values ( B± >0
are equal to zero, and ( O’z(t) > obeys the closed
equation of motion (13). When ð 1 (at zero tem- perature) or when 6 % 2 (at finite temperature), this property still holds and equation (13) for C1 z( t) >
remains appropriate, provided nonetheless that the functions At(t) and A2(t) are finite, which imposes
6 > 0 [13]. When the values of 6 belong to this range, the average value of the particle position tends towards
where the relaxation functions are defined by equa-
tion (14) as time-integrals of the correlation functions of the operators Bt multiplied by phase factors. By using general properties of equilibrium correlation functions, one can easily show [5] that there exists
a detailed balance condition relating the numerator
and the denominator of equation (46); this condition
ensures that, independently of the precise value of 6
and of the value of the coupling, one gets U z( (0) > =
-
